• To assess student mastery, observe students during small-group work and workstations and engage them in teacher-student discussions
• A paper/pencil formative assessment example (M-3-2-3_Array Assessment and KEY.doc) may be used to help assess student level of understanding. There are multiple pages. Choose based upon your assessment needs.

“Today we are going to continue our work with multiplication. You will be given a problem to solve, and you will need to solve it in some way. We have been working on problems all year, but today you will be given new types of problems.”

Hold up the book My Full Moon Is Square by Elinor J. Pinczes so students can see the cover.

“Before we begin working with arrays I would like to read a book to you. Let’s look at the cover. What do you think the book will be about?”

Take student suggestions and predictions about the book.

“The title of the book is My Full Moon Is Square by Elinor J. Pinczes. As we read the story I want you to notice what is happening to the moon on each page.”

When you have finished reading the book, ask students questions about the book and the changes they saw in the moon. Sample questions could include:

• What did you learn from the book?
• Explain what you noticed about the moon.
• Why do you think the title says the moon is a square?
• How were the fireflies arranged?

“The fireflies were organized. They were all lined up in rows and columns. The fireflies were organized into what is called an array.”

Learning to use an array model for multiplication is important because it is linked to many mathematical concepts. The array model can be used to demonstrate algebraic properties such as the commutative property.

“An array is an arrangement of objects, pictures, boxes, etc. in rows and columns. An array can be used to represent a problem, or you can look at an array and figure out the total number of objects.”

“We have worked a lot with the grouping model in multiplication. Today we will continue our work but we will be working with another multiplication model. As a class we will be exploring the array model. We will also discuss how the array model is similar to and different from our grouping model. Let’s start a chart that will show the types of models we have used in our classroom for multiplication.” The following chart could be started prior to this lesson.

It is important that students understand the difference between rows and columns when discussing arrays. Remind students that rows go across and columns go up and down. Rows are always counted first, followed by columns.

Draw the following array of stars on the class chart. Include the following information in your discussion if students don’t already know about arrays.

An array has

• both rows and columns
• rows with the same number of objects, equally spaced
• columns with the same number of objects, equally spaced

Ask students to write a number sentence to describe their array. Remind students that rows are counted first, followed by columns. Write on the chart:

 “Now that we have talked a little more about the array model, does anyone have anything else to share about what you noticed in the book we read at the beginning of class?”

(Students might answer: “The fireflies—moon—made an array. It kept changing as the story went along.”) Ask students how the moon changed. Did they notice that the arrays in the books were only squares and the 3-by-5 model above is in the shape of a rectangle? The arrays in the book are called square arrays because the number of rows and columns are the same, and when we draw the array, it forms a square instead of a rectangle. That is why we call them square arrays.

1 × 1, 2 × 2, 3 × 3 …

Hand out copies of the Looking at Arrays worksheet (M-3-2-3_Looking at Arrays and KEY.doc).

Look at each array. Write down the number of rows and columns in each array. Circle the rows or columns. Practice writing the multiplication equation for each array. See if students notice that the last two arrays both equal 12 but look different. Explain the commutative property
4 × 3 = 3 × 4. The arrays look different but have the same number of objects for each equation.

Distribute 30 counters or cubes to each student, and have students place the counters on a corner of their desks. Students will be using the counters to make arrays based on the information you give them. As students are working, walk around to check their work.

Some prompts may be:

• Show me 3 rows of 2.
• Place 4 rows of 5 on your desk.
• Build a 6-by-3 array.
• Sam had 2 bags. Each bag had 5 apples. How would I represent this problem with an array model?
• Cindy had 4 plates of granola bars with 6 granola bars on each plate. How many granola bars did she have on all the plates? Use an array model to represent your answer.

Have students share their array models and discuss their thinking.

“I am going to give each of you a task to do on your own. While you are working, I will come around and visit with you. You will need to explain how you are solving the task. It is okay if you don’t solve it completely. You will need to write down your answer on a piece of paper. We will review the tasks as an entire class during the last ten minutes of class.”

Make sure all students have at least twelve counters.

“You have to find as many array models as you can that equal 12. I’m asking that you work alone on this task for the first few minutes. You will be able to share your work with a partner in a few minutes. I’ll let you know when to share your work. Don’t forget to record your findings.”

Walk around and observe students, making sure they understand.

Give students time to compare their work and recordings with a partner. At the end of the class, select students to share their work using the overhead, board, or some type of projection system (M-3-2-3_Arrays for 12 KEY.doc).

You will have opportunities to assess students while they are working on the task through discussions and questions. Students may need to be pulled into small groups to further clarify understanding. You may also assess student learning at another time.

Students will need many experiences with the multiplication problems to find the relationship between the different models used in multiplication and division.

Extension:

Use the activities and strategies listed below to meet the needs of your students during the year.

• Routine: Show students a picture with groups of fruit arrays (M-3-2-3_Fruit Array.doc). Have students verbalize how many are in each group, and how many there are in all. Students might say:

“The oranges make a 3-by-2 array. That makes a total of six oranges.”

“Three groups of two oranges is the same as six oranges.”

The picture can be changed during the year to fit students’ interests.

• Mini-Lesson: Pose a multiplication problem to students:

“Stanley placed some hamburgers on a grill. He put the hamburgers in 3 rows with 5 in each row. How many hamburgers were on the grill?”

Have students build an array with circle counters. Discuss the arrays that were built. Some students will not see that a 3-by-5 array is the same as a 5-by-3 array. Place the arrays on top of each other so students can see this similarity. Lead the discussion toward the commutative property. Repeat the process multiple times.

• Small Group: Give each student a laminated Number Card Mat (M-3-2-3_Number Card Mat.doc). Shuffle two sets of the Number Cards (M-3-2-3_Number Cards.doc) and place them in a pile. Have students draw two cards from the pile and place them on their mat. Tell each student to make an array based on the two number cards. Once the first array is completed, ask the student to switch the cards and make a second array.

You also may have all students use the same numbers. Assess how students are solving the problem and clarify any misunderstandings.

• Workstation: Set up workstations around the room or divide students into small groups. Each player needs a recording sheet (M-3-2-3_Spin and Spin Again Recording Sheet.doc). Each group needs a paper clip or spinner component and both Spin and Spin Again spinners (M-3-2-3_Spin and Spin Again Spinners.doc). Players spin both spinners and record the multiplication sentence on their recording sheet. Players must also draw a dot array for each multiplication sentence.

Players continue taking turns until each player has written five multiplication sentences and drawn the arrays. All players find the total sum of their products. The player with the greatest sum is the winner. (To change numbers in the spinners, see M-3-2-3_Spin and Spin Again Spinner Template.doc.)

These lessons are designed as a progression from learning about relative size of a number to estimating large numbers of items to estimation culminating in hands-on activities and the use of visual references.